Global smooth solution to the 2D Boussinesq equations with fractional dissipation

TL;DR

This paper proves the global existence of smooth solutions for the 2D Boussinesq equations with fractional dissipation under specific conditions on the fractional powers, extending understanding of the system's regularity.

Contribution

It establishes new conditions on fractional dissipation parameters ensuring global smooth solutions for the 2D Boussinesq system, improving previous regularity results.

Findings

Global smooth solutions exist under certain fractional dissipation conditions.

Derived explicit bounds on fractional powers for solution regularity.

Extended the range of parameters for which solutions remain smooth.

Abstract

In this paper, we consider the two-dimensional (2D) incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. Based on the previous works and some new observations, we show that the condition $1-\alpha <\beta<\min\Big\{3-3\alpha,\,\,\frac{\alpha}{2},\,\, \frac{3\alpha^{2}+4\alpha-4}{8(1-\alpha)}\Big\}$ with $0.7351\approx\frac{10-2\sqrt{10}}{5}<\alpha<1$ suffices in order for the solution pair of velocity and temperature to remain smooth for all time.